Heat transmission in conjugate systems (cubes, cylinders, spheres) with travelling boundary line is investigated on the basis of the heat-conductivity nonlinear equation with a heat source obtained from the solution of an electrodynamic problem in a nonlinear medium whose dielectric penetrability depends on the field according to the law $$ \varepsilon_i=\varepsilon_{0i}-|\alpha_i|E_i^2,\qquad i=1,2. $$ The analysis carried out points out some nontrivial effects accompanying the heat transmission: the appearance of adiabatic surfaces, emergence of solition solutions and the conditions with sharpening in self-focusing media. All the considered effects are essentially defined by the values of $\alpha_i$ nonlinear parameters, nature of dependences of dielectric penetrabilities on the temperature, and the character of movement of the boundary surface.